Finally, recall that because the spin\(^c\)  structure is torsion, there is a natural absolute \(\mathbb {Q}\) -grading \(\mathrm {gr}^{\mathbb {Q}}\)  on Floer homology [1]}. Denoting by \(d\)  the absolute \(\mathbb {Q}\) -grading of the bottom of the \(U\) -tower of \(\) HM\(\) 

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\(\widehat{\phantom{\mathit {HM}}}\) 
 HM
HM(Y,s)\(, the Frøyshov invariant is defined to be \) h(Y,s)=-d/2\( \cite [Chapter 39]{KM}. This is an important quantity that provides information about the negative definite manifolds bounded by \) (Y,s)\(. We have the following.\begin{prop}The Frøyshov invariant h(Y,\mathfrak {s}) satisfies the inequality\begin{equation*}|h(Y,\mathfrak {s})|\le \frac{1}{2}\left(|gr^{\mathbb {Q}}([B_{\bullet }])|+\max |\mathsf {sf}(\widehat{\mathrm {Hess}}\mathcal {L}_{(B_{\bullet },0)},\widehat{\mathrm {Hess}}_{\mathcal {L}(B,\Psi )})|+1\right).\end{equation*}\end{prop}{\begin{xmlelement*}{proof}For every irreducible critical point x=[B,\Psi ], we have\begin{equation*}\mathrm {gr}^{\mathbb {Q}}(x)=\mathrm {gr}^{\mathbb {Q}}([B_{\bullet }])+\mathrm {gr}(x,[B_{\bullet }])=\mathrm {gr}^{\mathbb {Q}}([B_{\bullet }])+\mathsf {sf}(\widehat{\mathrm {Hess}}\mathcal {L}_{(B_{\bullet },0)},\widehat{\mathrm {Hess}}_{\mathcal {L}(B,\Psi )}).\end{equation*}Hence, the stable reducible critical point [\mathfrak {b}_i] with minimum degree larger than\begin{equation*}M=\mathrm {gr}^{\mathbb {Q}}([B_{\bullet }])+\max \mathsf {sf}(\widehat{\mathrm {Hess}}\mathcal {L}_{(B_{\bullet },0)},\widehat{\mathrm {Hess}}_{\mathcal {L}(B,\Psi )})\end{equation*}is not the differential of any irreducible element for degree reasons, and is a cycle as differentials from boundary stable to irreducible critical points involve two-step trajectories involving boundary unstable critical points, which are also not present for degree reasons. Hence [\mathfrak {b}_i] determines a non-zero class in the U-tower. Finally, there are no generators in the Floer chain complex in degrees less than\begin{equation*}m=\mathrm {gr}^{\mathbb {Q}}([B_{\bullet }])+\min \mathsf {sf}(\widehat{\mathrm {Hess}}\mathcal {L}_{(B_{\bullet },0)},\widehat{\mathrm {Hess}}_{\mathcal {L}(B,\Psi )})\end{equation*}and the result is proved.\end{xmlelement*}}\begin{comment}\begin{remark}When the Dirac operator D_{B_{\bullet }} has no kernel (which is part of the transversality assumptions), gr^{\mathbb {Q}}([B_{\bullet }]) can be expressed in terms of the \eta  invariants of the operators \ast d on coexact 1-forms and D_{B_{\bullet }}; this was studied in detail in \cite {LL2} in the case of hyperbolic rational homology spheres.\end{remark}\end{comment}\)
